We started this semester with a problem that Archimedes worked on (circa 250 BC) - estimating the area of a circlebefore we knew the formula A = (pi) R^2. There were some imaginative variations in how people tried to estimate the area - some were quite artistic! After looking at different approahes we noted these common themes: (1) we need to use more and more shapes that are individually getting smaller and smaller in order to get closer and closer to the exact area (2) limits seem to be involved in this problem! We then listed the steps of the "Integration Process" (1) Break into pieces (2) approximate each piece (3) ad up the approximations and find the limit of this sum as the number of approximations goes to infinity (and the size of each individual approximation approaches zero) We then looked at Riemann sums, which for now we described as a rectangular estimation method for estimating areas of regions that are bounded by curves in an x-y coordinate system.
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We started this semester with a problem that Archimedes worked on (circa 250 BC) - estimating the area of a circlebefore we knew the formula A = (pi) R^2. There were some imaginative variations in how people tried to estimate the area - some were quite artistic! After looking at different approahes we noted these common themes:
(1) we need to use more and more shapes that are individually getting smaller and smaller in order to get closer and closer to the exact area
(2) limits seem to be involved in this problem!
We then listed the steps of the "Integration Process"
(1) Break into pieces
(2) approximate each piece
(3) ad up the approximations and find the limit of this sum as the number of approximations goes to infinity (and the size of each individual approximation approaches zero)
We then looked at Riemann sums, which for now we described as a rectangular estimation method for estimating areas of regions that are bounded by curves in an x-y coordinate system.
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